Subgroups — #14546366

Order 2187.

173 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624754	C₃	1	2	yes	maximal, complemented
2	#10624505	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624449	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
5	#6911	—	3	2	no	—
6	#6910	—	3	2	no	—
7	#6909	—	3	2	no	—
8	C₃×AΣL(1,𝔽₂₇)	—	3	2	no	—
9	C₃×C₃²⋊C₉	—	3	1	no	—
10	#6866	—	3	1	no	—
11	C₃⁴	—	3	1	no	—
12	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
13	C₃×3₋¹⁺²	—	3	2	no	—
14	C₃×3₋¹⁺²	—	3	2	no	—
15	C₃×H(3,𝔽₃)	—	3	2	no	—
16	C₃×H(3,𝔽₃)	—	3	2	no	—
17	C₃×H(3,𝔽₃)	—	3	1	no	—
18	C₃×H(3,𝔽₃)	—	9	3	no	—
19	C₃²×C₉	—	9	1	no	—
20	#732	—	9	6	no	—
21	#731	—	9	6	no	—
22	AΣL(1,𝔽₂₇)	—	9	6	no	—
23	AΣL(1,𝔽₂₇)	—	9	2	no	—
24	AΣL(1,𝔽₂₇)	—	9	2	no	—
25	AΣL(1,𝔽₂₇)	—	9	2	no	—
26	C₃²⋊C₉	—	9	3	no	—
27	C₃²⋊C₉	—	9	1	no	—
28	C₃³	—	9	1	no	—
29	C₃³	—	9	1	no	—
30	C₃³	—	9	1	no	—
31	C₃³	—	9	1	no	—
32	C₃³	—	9	3	no	—
33	C₃³	—	9	2	no	—
34	C₃³	—	9	1	no	—
35	C₃³	—	27	1	no	—
36	C₃³	—	27	1	no	—
37	C₃³	—	27	1	no	—
38	C₃³	AΣL(1,𝔽₂₇)	1	2	yes	—
39	C₃³	#732	1	1	yes	characteristic
40	C₃³	#732	1	1	yes	characteristic
41	3₋¹⁺²	—	9	4	no	—
42	3₋¹⁺²	—	9	4	no	—
43	3₋¹⁺²	—	9	2	no	—
44	3₋¹⁺²	—	9	2	no	—
45	H(3,𝔽₃)	—	3	6	no	—
46	H(3,𝔽₃)	—	9	3	no	—
47	H(3,𝔽₃)	—	9	4	no	—
48	H(3,𝔽₃)	—	9	4	no	—
49	H(3,𝔽₃)	—	9	2	no	—
50	C₃×C₉	—	9	3	no	—
51	C₃×C₉	—	9	2	no	—
52	C₃×C₉	—	9	2	no	—
53	C₃×C₉	—	27	2	no	—
54	C₃×C₉	—	27	1	no	—
55	C₃×C₉	—	27	1	no	—
56	C₃²	—	3	1	no	—
57	C₃²	—	3	1	no	—
58	C₃²	—	3	1	no	—
59	C₃²	—	3	1	no	—
60	C₃²	—	3	2	no	—
61	C₃²	—	3	1	no	—
62	C₃²	—	9	2	no	—
63	C₃²	—	9	1	no	—
64	C₃²	—	9	3	no	—
65	C₃²	—	27	1	no	—
66	C₃²	—	27	1	no	—
67	C₃²	—	27	1	no	—
68	C₃²	—	27	3	no	—
69	C₃²	—	27	3	no	—
70	C₃²	—	27	3	no	—
71	C₃²	—	27	3	no	—
72	C₃²	—	27	2	no	—
73	C₃²	—	81	2	no	—
74	C₃²	#6856	1	1	yes	characteristic
75	C₉	—	27	3	no	—
76	C₉	—	27	2	no	—
77	C₉	—	27	2	no	—
78	C₉	—	81	1	no	—
79	C₃	—	3	1	no	—
80	C₃	—	3	1	no	—
81	C₃	—	3	1	no	—
82	C₃	—	3	1	no	—
83	C₃	—	9	1	no	—
84	C₃	—	9	2	no	—
85	C₃	—	27	3	no	—
86	C₃	—	81	2	no	—
87	C₃	#10624474	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃	#10624474
commutator, Frattini, cosocle	C₃²×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃²×H(3,𝔽₃) ▷ C₃⁴ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃⁴ ◁ C₃²×H(3,𝔽₃) ◁ G
LECS	G ▷ C₃²×H(3,𝔽₃) ▷ C₃⁴ ▷ C₃² ▷ C₃ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #10624754	C₃	semidirect
2. #10624505	C₃	semidirect
3. #10624449	C₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#10624754	3
#10624505	3
#10624449	3